Note that by definition, \(\mathsf {V}X^\mathbb {2}\subseteq \mathsf {V}X\times \mathsf {V}X\)  is precisely the graph of the partial order on \(X\) . It follows that \(R_T\)  is simply the lifting of the partial order on \(X\)  by the functor \(T\) (see [1]}), often denoted \(\overline{T}\hspace{-3.0pt}\le \) . In the rest of this section we will see examples where \(R_T\)  is transitive, where \(R_T\)  is transitive and antisymmetric, and where \(R_T\)  is not even transitive.
